ADDITION AND RESOLUTION OF VECTORS
11
dent, the components of r along any one of the axes must equal the sum of the corresponding components of the given vectors. Therefore (3) can be split into the following three separate equations.
rtf = ay+btf+ctf + • - • , (4)
rz = a, + b, + c, + ••-..
It was shown in § 11 that when two vectors are parallel the algebraic sum of their magnitudes equals the magni-
Yl
Fro. 12.
tude of their resultant. This result may be extended to any number of parallel vectors. Therefore we can put the vector equations of (4) into the following algebraic forms:
rv- av+lv+cy+
(5)
Equations (5) determine r through the following relations:
(6) (7)
9" rrr
rx
COS oti =
COS C*2 ^ ........J
cos «3=
where «i, «2, and «3 are the angles r makes with the axes.
16. Multiplication and Division of a Vector by a Scalar.— When a vector is multiplied or divided by a scalar the result is a vector which has the same direction as the original vector. If, in the equation b = ma, m be a scalar then b has the same direction as a but its magnitude is m times that of a.